Neither shoes nor sandals may be worn on Yom Kippur. In the past, these were generally made of leather, because people did not know how to craft strong, durable, and flexible shoes from other materials. Shoes made of cork, rubber, or wood were often used at home, as slippers, and poor people, who would normally go barefoot, would sometimes wear them outdoors. The question arises: May one wear non-leather footwear on Yom Kippur?
Some Rishonim forbid walking in wooden shoes, because one walking in them does not feel the roughness of the ground beneath his feet. However, they permit cork and rubber shoes, because one walking in them feels the roughness of the ground and suffers accordingly (Rashi; Rambam; Tosafot; Rabbeinu Yeruḥam).
Other Rishonim permit wearing all non-leather shoes. They maintain that, by definition, non-leather footwear cannot be shoes. Rather, such “shoes” are just items of clothing, and as such they may be worn on Yom Kippur (Ramban; Rosh; Rashba). Indeed, most Aḥaronim rule this way in practice (SA 614:2).
However, it seems clear that this view presumes a reality in which non-leather shoes were uncomfortable for walking, and thus it could be claimed that these were not considered proper shoes. Nowadays, however, when manufacturers commonly produce high-quality non-leather shoes, one may not wear shoes of any material on Yom Kippur if it is a kind of shoe that people would wear year-round to walk on rocky and rough terrain.
A generation ago, when it was still uncommon to find high-quality shoes made from other materials, some poskim permitted walking in comfortable shoes as long as they were not made from leather or synthetic leather. However, with the passage of time, excellent non-leather shoes are becoming more and more readily available, so the numbers of those who permit wearing such shoes on Yom Kippur are decreasing.
Therefore, one may not wear non-leather shoes on Yom Kippur if they are worn year-round on rocky or rough terrain, regardless of what material they are made from. Thus, footwear such as “Crocs,” “Keds,” and “All-stars” may not be worn on Yom Kippur. One may wear cloth slippers or basic rubber shoes, however, since they are not normally worn on rough terrain. Nevertheless, since some poskim are still permissive and permit non-leather shoes, one should not object if someone else relies on them.
It seems clear, however, that those who were lenient in the past assumed that no material could compete with leather in terms of durability, strength, and flexibility. All the substitutes were either cloth used as slippers or extremely low-quality shoes that only the poor would wear sometimes, as the poor usually went barefoot. They would wrap their feet for protection only if they had injured a foot or if they had to traverse particularly rough terrain. It is this type of “shoe” that the Amora’im and Rishonim argued about. Those who were stringent forbade wearing these shoes since they did offer a degree of protection. Those who were lenient permitted them because they were not comfortable to walk in and people did not generally do so. For example, Ritva on Shabbat 66a writes explicitly that he permitted wooden shoes because they were not normally worn. Ran, too, (Yoma 2b) offers this as the reason for the lenient position. Other Rishonim state that any shoe worn throughout the year may not be worn on Yom Kippur (Yere’im §420; Tosafot, Yevamot 103a, s.v. “be-anpilya”). Maharshag writes that we rule leniently when it comes to non-leather shoes, because quality shoes that people regularly wear are usually made of leather. Therefore, even in the rare case of good shoes that are made of another material, they may be worn (Responsa Maharshag 2:110). However, now that shoes are made from a variety of materials, Maharshag, too, would be stringent.
To summarize, it seems to me that there would be no disagreement among the Rishonim about shoes nowadays; all would agree that if the shoes are good quality and worn year-round, they may not be worn on Yom Kippur. Indeed, this is the ruling of R. Ariel (Ohala Shel Torah 2:81) and R. Elyashiv (cited in Hilkhot Ḥag Be-ḥag 22:25). It seems that as time goes on and people get more used to wearing shoes made from a variety of materials, more poskim are stringent and consider all of them as shoes. The law as it applies to flip-flops, Crocs, and the like is a bit unclear. On the one hand, many people do wear them in the street. However, it seems that the criterion to determine whether they are prohibited is whether people wear them on rough terrain. If almost no one wears them in such areas, they may be worn on Yom Kippur. (True, according to Baḥ one must walk barefoot, but we do not take his opinion into account.)
Perhaps we can justify the position of those who are lenient and wear good non-leather shoes. They may be following the view of the Rishonim who maintain that the prohibition on wearing shoes is a rabbinic enactment limited to leather. So even if nowadays all would agree that a non-leather shoe is good and sturdy, it does not become prohibited, as we do not make new enactments. Additionally, shoes made of leather are still considered to be of a higher quality than those made of any other material. Finally, Arizal offers a kabbalistic explanation for the prohibition, related to the “garments of leather” that God gave to Adam and Ḥava to appease their animal souls (Pri Etz Ḥayim, Sha’ar Yom Ha-kippurim, ch. 4). This rationale would not apply to non-leather shoes. Nevertheless, it seems to me that there is no room for leniency, and the prohibition applies to all shoes worn outside on rough terrain, whether or not they are leather.